Fluctuation scaling in complex systems: Taylor's law and beyond1

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form ‘fluctuations ≈ constant × average^α’, where the exponent α is predominantly in the range [1/2, 1]. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names Taylor's law or fluctuation scaling. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling.     

单色仪的定标实验中汞光谱两条谱线的补充标定

讨论了普通物理光学实验有关教材中单色仪定标实验中定标所依据的汞光谱谱线标定问题，通过实验确定了实验可以明显观察到而未能标定的谱线，对原有教材有关内容给出了必要的补充。     

Anomalous scaling in systems partially controlled by diffusion

We study the nature of anomalous scaling in several systems partially controlled by diffusion. We quantify the departure from Fickian scaling by means of an apparent exponent governing the scaling of long-time behavior with system size. We find that anomalous scaling should be expected whenever complex geometries, higher dimensionality, or time-dependent boundary conditions are encountered.     

Bethe ansatz calculations for the eight-vertex model on a finite strip

Bethe ansatz equations for the eigenvalues of the transfer matrix of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to be in agreement with recent studies of theXXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data.     

The Scaling Limit Geometry of Near-Critical 2D Percolation

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = p _c+λδ^1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = p _c, based on SLE ₆. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of “macroscopically pivotal” lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.     

Lagrangian Dispersion in Gaussian Self-Similar Velocity Ensembles

We analyze the Lagrangian flow in a family of simple Gaussian scale-invariant velocity ensembles that exhibit both spatial roughness and temporal correlations. We argue that the behavior of the Lagrangian dispersion of pairs of fluid particles in such models is determined by the scale dependence of the ratio between the correlation time of velocity differences and the eddy turnover time. For a non-trivial scale dependence, the asymptotic regimes of the dispersion at small and large scales are described by the models with either rapidly decorrelating or frozen velocities. In contrast to the decorrelated case, known as the Kraichnan model and exhibiting Lagrangian flows with deterministic or stochastic trajectories, fast separating or trapped together, the frozen model is poorly understood. We examine the pair dispersion behavior in its simplest, one-dimensional version, reinforcing analytic arguments by numerical analysis. The collected information about the pair dispersion statistics in the limiting models allows to partially predict the extent of different phases of the Lagrangian flow in the model with time-correlated velocities.     

Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and −1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x=0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L ^1.78(2). Anomalous scaling is not found for the characteristic energy, which essentially scales as L ². When x=0.25, a crossover to L ² scaling of the entropy is seen near L=12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L. PACS numbers: 75.10.Nr, 75.40.Mg, 75.50.Lk     

A universal scaling law for hadronic reactions

A master scaling law is proposed for arbitrary distributions in arbitrary hadronic processes of which all experimentally established scaling laws (and a host of others, easily deduced as occasion demands) are special cases.     

Scaling Region in Desynchronous Coupled Chaotic Maps

The largest Lyapunov exponent and the Lyapunov spectrum of a coupled map lattice are studied when the system state is desynchronous chaos. In the large system size limit a scaling region is found in the parameter space where the largest Lyapunov exponent is independent of the system size and the coupling strength. Some scaling relation between the Lyapunov spectrum distributions for different coupling strengths is found when the coupling strengths are taken in the scaling parameter region. The existence of the scaling domain and the scaling relation of Lyapunov spectra there are heuristically explained.     

Scaling Region in Desynchronous Coupled Chaotic Maps

The largest Lyapunov exponent and the Lyapunov spectrum of a coupled map lattice are studied when the system state is desynchronous chaos. In the large system size limit a scaling region is found in the parameter space where the largest Lyapunov exponent is independent of the system size and the coupling strength. Some scaling relation between the Lyapunov spectrum distributions for different coupling strengths is found when the coupling strengths are taken in the scaling parameter region. The existence of the scaling domain and the scaling relation of Lyapunov spectra there are heuristically explained.     

Universal characteristic function of an adsorption-desorption process

The characteristic function of the initial order for the adsorption-desorption process is obtained by Monte Carlo simulation. The universality of the generalized scaling relation is confirmed by use of the sequential and synchronous dynamic rules. In the off-critical regime, we observe a "finger-print effect"of the initial state.     

Numerical experiments on billiards

We investigate decay properties of correlation functions in a class of chaotic billiards. First we consider the statistics of Poincaré recurrences (induced by a partition of the billiard): the results are in agreement with theoretical bounds by Bunimovich, Sinai, and Bleher, and are consistent with a purely exponential decay of correlations out of marginality. We then turn to the analysis of the velocity-velocity correlation function: except for intermittent situations, the decay is purely exponential, and the decay rates scale in a simple way with the (uniform) curvature of the dispersing arcs. A power-law decay is instead observed when the system is equivalent to an infinite-horizon Lorentz gas. Comments are given on the behaviour of other types of correlation functions, whose decay, during the observed time scale, appears slower than exponential.     

波数线性校正方法在多烯烃和类胡萝卜素Raman光谱频率校正中的应用

采用密度泛函理论计算了两种类胡萝卜素(β-胡萝卜素和叶黄素) 和几种短链(n=2~5)多烯烃的Raman光谱,用波数线性校正(WLS)方法对计算频率进行了校正,并与常用的几种校正方法作了比较。最常用的单一参数校正(UFS)方法只适用于个别频率的校正,对于所有振动频率的综合校正效果并不理想。WLS方法对于多烯烃和类胡萝卜素的校正结果明显优于UFS方法,校正公式分别为ν_obs／ν_calc=0.999 9-0.000 027 4ν_calc和ν_obs／ν_calc=0.993 8-0.000 024 8ν_calc,这些结果说明WLS方法可以用于类胡萝卜素这样的大分子的频率校正。WLS方法对多烯烃和类胡萝卜素的校正参数非常接近,证明WLS方法对多烯烃的校正结果可以用于类胡萝卜素的频率校正,这为类胡萝卜素频率校正提供了一种新的方法。     

Universality of 3D percolation exponents and first-order corrections to scaling for conductivity exponents

By using a fast, Nested Dissection algorithm we compare the results of finite-size scaling at p_c and of “p” scaling () on large cubic random resistor networks [up to 500×500×500]. The “p” scaling for conductivity of both site and bond networks leads to an exponent t=2.00(1). The finite-size scaling leads to the ratio of this conductivity exponent to the coherence length exponent ν: t/ν=2.283(3). Combining these results we estimate ν=0.876(6), in excellent agreement with a value proposed by Ballesteros et al. The first-order correctional exponent ω is found to be ω=1.0(2).     

The Finite-Size Scaling Functions of the Four-Dimensional Ising Model

A finite-size scaling function of the Privman–Fisher form is proposed for the singular part of the free-energy density of the four-dimensional Ising model. It leads to the finite-size scaling relations available and to the prediction of new ones.     

Cluster size distribution at criticality

Monte Carlo studies of the cluster size distribution for the site percolation problem on the triangular lattice are extended to lattices with up to 4 × 10¹¹ sites. Agreement with the predictions of scaling theory at p_c is excellent over a range of cluster sizes spanning five orders of magnitude.     

Extensive numerical study of the anomalous dynamic scaling of the Wolf-Villain model

In order to study the microscopic physical mechanisms of roughness surfaces exhibiting the anomalous scaling behavior, the Wolf-Villain model in 1+1 and 2+1 dimensions is investigated by the kinetic Monte-Carlo simulation on long time and large length scale (the growth time and the system size are respectively extended to t=2²⁹, for 1+1 dimensions, and t=2²¹, L×L=512×512 for 2+1 dimensions). In the 2+1-dimensional simulations, the noise reduction technique is employed so as to eliminate the crossover effects in the growth process. Our calculations show that the Wolf-Villain model in 1+1 dimensions very probably exhibits intrinsic anomalous scaling behavior in the time and length simulation range of this paper, and the 2+1-dimensional Wolf-Villain model leads to a pyramidal mounded morphology. Some properties of the mounded pattern in the 2+1-dimensional Wolf-Villain model are discussed in the final part of this presentation.     

Steady-state scaling function of the (1 + 1)-dimensional single-step model

The steady-state height-height correlation function for the (1 + 1)-dimensional single-step model is calculated in a large-scale Monte Carlo simulation. Analysis of the data yields a universal ratio of scaling amplitudes which differs from the value obtained recently from a mode-coupling calculation. An empirical form for a universal scaling function is also presented.     

Importance of packing in spiral defect chaos

We develop two measures to characterize the geometry of patterns exhibited by the state of spiral defect chaos, a weakly turbulent regime of Rayleigh-Bénard convection. These describe the packing of contiguous stripes within the pattern by quantifying their length and nearest-neighbor distributions. The distributions evolve towards a unique distribution with increasing Rayleigh number that suggests power-law scaling for the dynamics in the limit of infinite system size. The techniques are generally applicable to patterns that are reducible to a binary representation.